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Questions tagged [topological-groups]

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

A topological group is a group endowed with a topology such that both the group operation and inversion are continuous maps. Every group can be understood as a topological group, if we take the discrete topology.

Topological groups are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis, see e.g. Pontryagin duality.

2270 questions
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The inverses of open sets

I am not sure if this is already posted, though, I hope I can get some help, and thank in advance. This question arises from the proof of the following. Proposition: Let G be a topological group, of which H is a subgroup. Then, H is closed in G,…
awllower
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Topological group: if $g_n, h_n, k_n \in G$ s.t. $g_n h_n \to e$ and $k_n \to e$ does $g_n k_n h_n \to e$?

This is certainly true if $g_n$ (equivalently $h_n$) converge in $G$ and it feels like it should be true but after trying to prove it, it's unclear.
someone
  • 160
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Strengthening the second isomorphism theorem for topological groups

In a previous question I asked about the following version of the second isomorphism theorem for topological groups -I implicitly assume that every subgroup inherits the subspace topology-: Let $G$ be a topological group and $H$ and $N$ subgroups.…
Josué Tonelli-Cueto
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Proving that $f:G\to G $ such that $f(x)=a\cdot x$ is an homeomorphism

I'm having some trouble with the following exercise: Let $(G,\cdot)$ be a topological group, and let $a\in G$. Consider the function: $$f:G\to G \\ \ \ \ \ \ \ \ \ \ \ x\mapsto a\cdot x$$ Prove that $f$ is an homeomorphism I was easily able to…
Eduardo Magalhães
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I want to prove this is a group topology

We have an epimorphism $f:G\to H$ between topological groups. We consider $\tau$ as the family of all images $f(U)$ where $U\subseteq G$ is open in $G$. The assertion is that $\tau$ is a topology over $H$ such that $(H,\tau )$ is a topological…
Talexius
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Inverse limit of locally compact groups

If $(G_i,\varphi_i)$ is a inverse system of Hausdorff locally compact groups, then $\varprojlim G_i $ is locally compact? I have looked for information about it, or any additional condition for the conclusion holds. I don't know if I added the…
Luis Antonio Sanchez
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Question about $\langle b\rangle$ in a Hausdorff Topological Group

Let $G$ be a topological Hausdorff group, if $b\in G$, does there exist an open neighborhood of $b$ such that $U\cap \langle b\rangle$ is finite. I know this is a really odd question, but I would really appreciate to know whether this is true or…
Camilo Arosemena-Serrato
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The topology of the free topological group

I don't understand something about the following result: Claim: The topology of the group $F(X)$ is the finest topological group topology on $F_a(X)$ that generates on $X$ its original topology. ($X$ is a Tychonoff topological space, and $F(X)$…
user73564
  • 569
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If $H \le K \le G$ then $H \trianglelefteq \overline{K}$?

If I have topological groups $H \le K \le G$, I could prove that if $H \trianglelefteq K$ then $\overline{H} \trianglelefteq K$ but is it also true that $H \trianglelefteq \overline{K}$?
roi_saumon
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Is a continuous map between two topological groups homotopic to a homomorphism between them?

Let $G$ and $H$ be two topological groups and $f:G\to H$ be a continuous map. Is there a continuous homomorphism $g:G\to H$ homotopic to $f$?
M.Ramana
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The interval [0,1] with additional modulo 1

The interval $[0, 1]$ is an abelian group with addition modulo $1$. Let $H$ be a proper subgroup of $[0, 1]$, which is closed as subset of $[0,1]$. Show $H$ is finite. I assumed $H$ is infinite: since $H$ is closed, all limit points of $H$ are in…
Zikzak
  • 31
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Extension of a finite group by a connected group necessarily splits?

Suppose that $G$ is a compact abelian group. I denote by $G_0$ the connected component of the identity in $G$. If $G_0$ is open in $G$ (equivalently $G/G_0$ is finite) is it true that $G\cong G_0\times G/G_0$? I also assume that $G/G_0$ is equipped…
Yanko
  • 13,758
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Argument that if $H$ is normal subgroup of $G$, then $G/H$ is topological group.

Let $G$ be a topological group and $H$ a normal subgroup of $G$. We want to prove that $G/H$ is a topological group. I have found this argument: We know that canonical projection $\rho\colon G \rightarrow G/H$ is an open map. We also have the…
HeMan
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Action and bounded orbits

Let $H$ be an open group such that, $H$ act continously an by isometrieson a metric space $(X,d)$ ($\forall h\in H$, the map $X\ni x\longmapsto h.x\in X$ is an isometry.). Recall that for $x_{0}\in X$ the orbit of $x_{0}$ is the following set…
Serges
  • 479
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Is there a topological version of Cayley's Theorem?

Given a topological group $G$ can we embbed it in $Aut_{TOP}(X)$ for some topological space $X$? It is a good choice the open-compact topology for $Aut_{TOP}(X)$?
Math.mx
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